Vector addition & magnitude (video) | Khan Academy

Video transcript

Sal:Let’s say that we have three vectors, vectors A, B and C, and we know that vector A plus vector B
is equal to vector C. Now given this I have some
interesting questions. Can you construct a scenario where the magnitude of vector C is equal to the magnitude of vector A plus
the magnitude of vector B? And can you also, using potentially
different vectors A and B, construct a scenario where the magnitude of vector C is greater than the magnitude of vector A plus
the magnitude of vector B? I encourage you to pause this video right now and try to do that. Try to come up with some
vectors A and B so that when you take their sum,
the magnitude of the sum is equal to the sum of the magnitudes. And also see if you can come
up with some vectors A and B so that if you take the sum of
the vectors that the magnitude of the sum is actually greater
than the sum of the magnitude. see if you can come up with that. I’m assuming you’ve given a go at it, and potentially you’ve gotten
a little bit frustrated, especially with the second one. The only way- Let’s actually
just draw some vectors. If you have vector A like this, and let’s say vector B
looks something like that, then A plus B … We can just shift this
over, copy and paste. A plus B is going to look like this. A plus B, or vector C I guess we could say, is going to look like that. And notice, these three
vectors always form a triangle. If you have a triangle, one side cannot be longer than the sum of
the other two sides. Think about it. If you wanted this to
be longer what you could try to do is maybe change
vector B in a way so you’re pushing it further and further out. Maybe if you change your
vector B a little bit you could get this vector C
to be longer and longer. Maybe if you made your vector B like this. Maybe your vector B would
look something like this. Now your vector C is getting
pretty long, but it’s still shorter than the
sum of these two sides. To make it equal to the sum of these two sides you essentially have to make these two vectors go in the
exact same direction. To make it equal you have to
have vector A looking like this. You need to change the direction of B, or essentially construct a vector B that’s going in the exact same direction. Only in this circumstance
will you get this scenario, where the magnitude of
vector C is equal to. Really the largest that
the magnitude of the sum can be is the sum of the
magnitudes, and that only happens when these two are
going in the same direction. These are going in the same,
in the exact same direction. This right over here is impossible. You could never get one side
of a triangle being longer than the sum of the other two sides,
based on what we just saw. You’re probably saying, “What
about the circumstance where “the magnitude of our sum is
less than the magnitude of …” Than the sum of the magnitudes,
I guess I could say. This is fairly typical. This is pretty much always the scenario, this is what’s always going to be the case when the vectors are not
in the same direction. If someone drew a vector like this- Let me draw that a little bit straighter. If someone drew a vector
like this and a vector like this these clearly are not
going in the same direction, so the sum of these two vectors, the magnitude of that is going to be less than the sum of these two magnitudes. For example if the
magnitude here is five and the magnitude here is three then we know that if we were to add
these two things … Let me just show you. Copy and paste. Actually let me just cut and paste, so that we can clean
things up a little bit. Cut and paste. Let’s add these two vectors. So we know that the sum of these two, which is going to be this
vector right over here, its magnitude is going to be
less than five plus three. It is going to be less than eight. The only way that this magnitude
could even get to eight is if these two vectors went
in the exact same direction.